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Integral Hopf-Galois structures on degree \(p^2\) extensions of \(p\)-adic fields. (English) Zbl 0992.11065

This paper gives the definitive classification of Hopf Galois structures on valuation rings of Galois extensions of \(p\)-adic fields of degree \(p^2\), \(p\) prime.
Let \(L/K\) be a finite, totally ramified Galois extension of \(p\)-adic fields of degree \(p^2\) with Galois group \(G\), and assume \(K\) contains a primitive \(p\)th root of unity. Let \(e\) be the absolute ramification index of \(K\). Then there are \(p\), respectively \(p^2\) Hopf Galois structures on \(L/K\) if \(G\) is cyclic, respectively elementary abelian. Their structure depends at most on a field extension \(M\) of degree \(p\) over \(K\). The author explicitly describes the corresponding \(K\)-Hopf algebras and their Hopf orders over \(\mathcal O_K\), the valuation ring of \(K\), extending work of C. Greither [Math. Z. 210, 37-68 (1992; Zbl 0737.11038)] and the reviewer [New York J. Math. 2, 86-102 (1996; Zbl 0884.11046)]. Using this he determines which of the Hopf orders are realizable, that is, can be the associated order of the valuation ring of a Galois extension of \(K\). The results depend on \(G\), two “valuation parameters”, \(i\) and \(j\), and a “unit parameter” \(v\) (in \(M\)), as well as whether or not \(p = 2\), and break up into seven cases corresponding to various subregions of the square \[ \{ (i,j) \mid 0 \leq i, j, \leq e/p \} . \] This is applied to determine which of the Hopf Galois structures on \(L\) give rise to Hopf Galois structures on \(\mathcal O_L\). The number of such Hopf Galois structures is either 1 or \(p\).
The paper concludes by fitting into this landscape two classes of examples: Galois field extensions \(L/K\) of degree \(p^2\) that are classical Kummer extensions, and Kummer extensions for Lubin-Tate formal groups. The latter examples extend work of M. J. Taylor [J. Reine Angew. Math. 358, 97-103 (1985; Zbl 0582.12008)] and the author [Galois module structure and Kummer theory for Lubin-Tate formal groups, in Algebraic Number Theory and Diophantine Analysis, ed. by F. Halter-Koch and R. F. Tichy, de Gruyter, Berlin 55-67 (2000; Zbl 0958.11076)].

MSC:

11S23 Integral representations
11R33 Integral representations related to algebraic numbers; Galois module structure of rings of integers
11S31 Class field theory; \(p\)-adic formal groups
16W30 Hopf algebras (associative rings and algebras) (MSC2000)
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